On a Study of the Totally Umbilical Semi-Invariant Submanifolds of Golden Riemannian Manifolds

The Golden Ratio is fascinating topic that continually generated news ideas. A Riemannian manifold endowed with a Golden Structure will be called a Golden Riemannian manifold. Precisely, we can say that an (1,1)-tensor field on a m-dimensional Riemann manifold is a Golden structure if it satisfies the equation , where is identity map on . Furthermore, , the Riemannian metric is called -compatible and is named a Golden Riemannian manifold. The main purpose of the present paper is to study the geometry of Riemannian manifolds endowed with Golden structures. For this purpose, we study totally umbilical semi-invariant submanifold of the Golden Riemannian manifolds. Also, we obtain integrability conditions of the distributions and investigate the geometry of foliations.

Anahtar Kelimeler:

Semi-invariant submanifolds, Golden Riemannian manifolds, totally umbilical submanifolds

On a Study of the Totally Umbilical Semi-Invariant Submanifolds of Golden Riemannian Manifolds

Keywords:

Semi-invariant submanifolds, Golden Riemannian manifolds, totally umbilical submanifolds,

PDF

___

[1] Bejancu, A., “Geometry of CR-submanifolds”, D. Reidel, (1986). [2] Chen, B.-Y. “Riemannian submanifolds, in Handbook of Differential Geometry”, North Holland (eds. F. Dillen and L. Verstraelen) I, 187418, (2000).
[3] Crasmareanu, M. and Hretcanu, C. E. “Golden differansiyel geometry”. Chaos, Solitons & Fractals 38, (5): 1229-1238, (2008).
[4] Crasmareanu, M. and Hretcanu, C. E. “Applications of the Golden Ratio on Riemannian Manifolds”. Turkish J. Math. 33, (2): 179-191, (2009).
[5] Gezer, A., Cengiz, N. and Salimov, A., “On integrability of Golden Riemannian structures”. Turkish J. Math. 37: 693-703, (2013).
[6] Hretcanu, C. E and Crasmareanu, M. “On some invariant submanifolds in a Riemannian manifold with golden structure”. An.Stiins. Univ. Al. I. Cuza Iasi. Mat. (N.S.)53, (1): 199-211, (2007).
[7] Sahin, B. and Akyol, M. A., “Golden maps between Golden Riemannian manifolds and constancy of certain maps”, Math. Commun. 19: 333-342, (2014).
[8] Yano, K. And Kon, M., “Structures on manifolds”, World Scientific, Singapuore, Series in pure mathematics, 3 (1984).
[9] Erdoğan, F. E. And Yıldırım C. "Semi-invariant submanifolds of Golden Riemannian manifolds" AIP Conference Proceedings, 1833, 020044, doi:10.1063/1.4981692, (2017).